I explored past posts about force calculations looking for a good basic equation to calculate forces on ropes, climbers, protection and so on. rgold's formula is great. Thanks!

My question involves the difference between calculating forces on climbers vs. protection. I believe the answer is related to the pulley effect and friction, but I'm not sure of the details. I hope somebody can help me see this clearly.

rgold's formula provides the force on the climber. To calculate the force on the protection, various posts suggest multiplying by 1.6. For example:

"The impact force calculated from this equation is the force on the climber. Multiply that by 1.6 to get the impact force on the piece of pro that held the fall."

or

"While we're on the subject, if you take account of friction between the rope and the top piece, then tension in the belayer's side of the rope is about 60% of the tension in the climber's side. The sum of these two forces is the force on the top piece. So, the top piece feels about 1.6 times the climber's weight, in a static situation. This ignores any other friction in the system, such as between the rope and intermediate protection or between the rope and the rock."

Can somebody explain the number 1.6 to me in more detail? Is this number based on an estimation of the friction between ropes and carabiners--or is it mathematically more precise than that?

Also, I can understand that because of friction the forces on the climber's and belayer's side are different, however, I would think that friction would add force to the protection, in the same way that friction adds force to an anchor in a hauling system, and you would have to multiply by more than 2 instead of 1.6.

"While we're on the subject, if you take account of friction between the rope and the top piece, then tension in the belayer's side of the rope is about 60% of the tension in the climber's side. The sum of these two forces is the force on the top piece. So, the top piece feels about 1.6 times the climber's weight, in a static situation. This ignores any other friction in the system, such as between the rope and intermediate protection or between the rope and the rock."

Can somebody explain the number 1.6 to me in more detail? Is this number based on an estimation of the friction between ropes and carabiners--or is it mathematically more precise than that?

My understanding is that it is an empirical result from experiments in which the tension in the belayer's side of the rope was found to be around 0.6 (or 0.67) times the tension in the climber's side. Since the total downward force on the anchor is the sum of these two tensions, the total force on the anchor is 1.6 (or 1.67) times the tension in the belayer's side of the rope (or, equivalently, the impact force on the climber).

"While we're on the subject, if you take account of friction between the rope and the top piece, then tension in the belayer's side of the rope is about 60% of the tension in the climber's side. The sum of these two forces is the force on the top piece. So, the top piece feels about 1.6 times the climber's weight, in a static situation. This ignores any other friction in the system, such as between the rope and intermediate protection or between the rope and the rock."

Can somebody explain the number 1.6 to me in more detail? Is this number based on an estimation of the friction between ropes and carabiners--or is it mathematically more precise than that?

My understanding is that it is an empirical result from experiments in which the tension in the belayer's side of the rope was found to be around 0.6 (or 0.67) times the tension in the climber's side. Since the total downward force on the anchor is the sum of these two tensions, the total force on the anchor is 1.6 (or 1.67) times the tension in the belayer's side of the rope (or, equivalently, the impact force on the climber).

Jay

This is the limiting case where the climber and belayer's ends of the rope are both parallel. As the angle deviates from this, the contribution of the belayer's side to the tension in the top piece will reduce. Counteracting this, however, is the fact that reduced contact area between rope and carabiner leads to reduced friction, increasing the tension in the belayer's side. So... it's complicated (read: more or less impossible) to calculate from first principles, and can only be estimated based on experiments.

"While we're on the subject, if you take account of friction between the rope and the top piece, then tension in the belayer's side of the rope is about 60% of the tension in the climber's side. The sum of these two forces is the force on the top piece. So, the top piece feels about 1.6 times the climber's weight, in a static situation. This ignores any other friction in the system, such as between the rope and intermediate protection or between the rope and the rock."

Can somebody explain the number 1.6 to me in more detail? Is this number based on an estimation of the friction between ropes and carabiners--or is it mathematically more precise than that?

My understanding is that it is an empirical result from experiments in which the tension in the belayer's side of the rope was found to be around 0.6 (or 0.67) times the tension in the climber's side. Since the total downward force on the anchor is the sum of these two tensions, the total force on the anchor is 1.6 (or 1.67) times the tension in the belayer's side of the rope (or, equivalently, the impact force on the climber).

Jay

This is the limiting case where the climber and belayer's ends of the rope are both parallel. As the angle deviates from this, the contribution of the belayer's side to the tension in the top piece will reduce. Counteracting this, however, is the fact that reduced contact area between rope and carabiner leads to reduced friction, increasing the tension in the belayer's side. So... it's complicated (read: more or less impossible) to calculate from first principles, and can only be estimated based on experiments.

The angle issue is minor enough that I think we can concentrate on a model in which both ropes are vertical. If you're going to get picky about rope angles, then you shouldn't consider the "parallel" case to be "limiting." The positioning of the belayer relative to the top anchor can result in a negative angle between the ropes at the top anchor (ie, the ropes cross) as easily as a positive angle.

"While we're on the subject, if you take account of friction between the rope and the top piece, then tension in the belayer's side of the rope is about 60% of the tension in the climber's side. The sum of these two forces is the force on the top piece. So, the top piece feels about 1.6 times the climber's weight, in a static situation. This ignores any other friction in the system, such as between the rope and intermediate protection or between the rope and the rock."

Can somebody explain the number 1.6 to me in more detail? Is this number based on an estimation of the friction between ropes and carabiners--or is it mathematically more precise than that?

My understanding is that it is an empirical result from experiments in which the tension in the belayer's side of the rope was found to be around 0.6 (or 0.67) times the tension in the climber's side. Since the total downward force on the anchor is the sum of these two tensions, the total force on the anchor is 1.6 (or 1.67) times the tension in the belayer's side of the rope (or, equivalently, the impact force on the climber).

Jay

This is the limiting case where the climber and belayer's ends of the rope are both parallel. As the angle deviates from this, the contribution of the belayer's side to the tension in the top piece will reduce. Counteracting this, however, is the fact that reduced contact area between rope and carabiner leads to reduced friction, increasing the tension in the belayer's side. So... it's complicated (read: more or less impossible) to calculate from first principles, and can only be estimated based on experiments.

The angle issue is minor enough that I think we can concentrate on a model in which both ropes are vertical. If you're going to get picky about rope angles, then you shouldn't consider the "parallel" case to be "limiting." The positioning of the belayer relative to the top anchor can result in a negative angle between the ropes at the top anchor (ie, the ropes cross) as easily as a positive angle.

Jay

Guess I wasn't clear enough. A bit more detail:

In the below force diagram, F is the force on the climber, and f is the frictional force through the carabiner, so that tension on the belayer side is F - f. Then as the angle changes, the vertical (Fv) and horizontal (Fh) components of the forces on the top carabiner are as shown. The total force on the top piece (Ft) is then calculated by good old Pythagoras.

Plugging in some numbers (setting F arbitrarily to 1, and f to 0.33 so that F - f = 0.67), we get the following for Ft vs. angle:

.

That's what I was getting at, anyway - that maximum force will be seen where belayer and climber are pulling in the same direction.

"While we're on the subject, if you take account of friction between the rope and the top piece, then tension in the belayer's side of the rope is about 60% of the tension in the climber's side. The sum of these two forces is the force on the top piece. So, the top piece feels about 1.6 times the climber's weight, in a static situation. This ignores any other friction in the system, such as between the rope and intermediate protection or between the rope and the rock."

Can somebody explain the number 1.6 to me in more detail? Is this number based on an estimation of the friction between ropes and carabiners--or is it mathematically more precise than that?

My understanding is that it is an empirical result from experiments in which the tension in the belayer's side of the rope was found to be around 0.6 (or 0.67) times the tension in the climber's side. Since the total downward force on the anchor is the sum of these two tensions, the total force on the anchor is 1.6 (or 1.67) times the tension in the belayer's side of the rope (or, equivalently, the impact force on the climber).

Jay

This is the limiting case where the climber and belayer's ends of the rope are both parallel. As the angle deviates from this, the contribution of the belayer's side to the tension in the top piece will reduce. Counteracting this, however, is the fact that reduced contact area between rope and carabiner leads to reduced friction, increasing the tension in the belayer's side. So... it's complicated (read: more or less impossible) to calculate from first principles, and can only be estimated based on experiments.

The angle issue is minor enough that I think we can concentrate on a model in which both ropes are vertical. If you're going to get picky about rope angles, then you shouldn't consider the "parallel" case to be "limiting." The positioning of the belayer relative to the top anchor can result in a negative angle between the ropes at the top anchor (ie, the ropes cross) as easily as a positive angle.

Jay

Guess I wasn't clear enough. A bit more detail:

In the below force diagram, F is the force on the climber, and f is the frictional force through the carabiner, so that tension on the belayer side is F - f. Then as the angle changes, the vertical (Fv) and horizontal (Fh) components of the forces on the top carabiner are as shown. The total force on the top piece (Ft) is then calculated by good old Pythagoras.

Plugging in some numbers (setting F arbitrarily to 1, and f to 0.33 so that F - f = 0.67), we get the following for Ft vs. angle:

.

That's what I was getting at, anyway - that maximum force will be seen where belayer and climber are pulling in the same direction.

My main point was just that we should ignore the affect of the angle, at least at this stage of answering the OP's question, and just assume that both ropes are vertical, or at whatever angle the experiments were conducted that came up with the 1/3 "friction factor."

My much more picayune point was that if you are going to concern yourself with the effect of the angle on friction, then you shouldn't ignore cases where the angle is less than zero, and hence the friction is greater (I think) than where both ropes are plumb.

And, BTW, wouldn't f itself be a function of theta, rather than a constant independent of theta?

"While we're on the subject, if you take account of friction between the rope and the top piece, then tension in the belayer's side of the rope is about 60% of the tension in the climber's side. The sum of these two forces is the force on the top piece. So, the top piece feels about 1.6 times the climber's weight, in a static situation. This ignores any other friction in the system, such as between the rope and intermediate protection or between the rope and the rock."

Can somebody explain the number 1.6 to me in more detail? Is this number based on an estimation of the friction between ropes and carabiners--or is it mathematically more precise than that?

My understanding is that it is an empirical result from experiments in which the tension in the belayer's side of the rope was found to be around 0.6 (or 0.67) times the tension in the climber's side. Since the total downward force on the anchor is the sum of these two tensions, the total force on the anchor is 1.6 (or 1.67) times the tension in the belayer's side of the rope (or, equivalently, the impact force on the climber).

Jay

This is the limiting case where the climber and belayer's ends of the rope are both parallel. As the angle deviates from this, the contribution of the belayer's side to the tension in the top piece will reduce. Counteracting this, however, is the fact that reduced contact area between rope and carabiner leads to reduced friction, increasing the tension in the belayer's side. So... it's complicated (read: more or less impossible) to calculate from first principles, and can only be estimated based on experiments.

The angle issue is minor enough that I think we can concentrate on a model in which both ropes are vertical. If you're going to get picky about rope angles, then you shouldn't consider the "parallel" case to be "limiting." The positioning of the belayer relative to the top anchor can result in a negative angle between the ropes at the top anchor (ie, the ropes cross) as easily as a positive angle.

Jay

Guess I wasn't clear enough. A bit more detail:

In the below force diagram, F is the force on the climber, and f is the frictional force through the carabiner, so that tension on the belayer side is F - f. Then as the angle changes, the vertical (Fv) and horizontal (Fh) components of the forces on the top carabiner are as shown. The total force on the top piece (Ft) is then calculated by good old Pythagoras.

Plugging in some numbers (setting F arbitrarily to 1, and f to 0.33 so that F - f = 0.67), we get the following for Ft vs. angle:

.

That's what I was getting at, anyway - that maximum force will be seen where belayer and climber are pulling in the same direction.

My main point was just that we should ignore the affect of the angle, at least at this stage of answering the OP's question, and just assume that both ropes are vertical, or at whatever angle the experiments were conducted that came up with the 1/3 "friction factor."

My much more picayune point was that if you are going to concern yourself with the effect of the angle on friction, then you shouldn't ignore cases where the angle is less than zero, and hence the friction is greater (I think) than where both ropes are plumb.

And, BTW, wouldn't f itself be a function of theta, rather than a constant independent of theta?

Jay

Indeed it would be - which is why, as I said, the only way to get good estimates of forces is by experiments with various configurations.

The value of 1.6 is roughly what you get on the top runner if you perform a static belay drop tests (in other words tie a knot in the rope) but through a belay plate you get rather different results. For something like a Grigri you might be getting up this high but for the rest of the time the factor will be considerably lower.

As jt says the force on the top piece is the sum of the two forces BUT this is not the sum of the two maximum forces, rather the sum of the two forces at the same time. (The belayer side force lags considerably behind that of the faller because of frictional hysterises and the top runner force is in-between.)

If you look at the force graphs for a typical device you will see that at the point of maximum force for the faller (3.08kN) the belayer side force is 0.8kN and the top runner force is 3.9kN which all adds up nicely. However the belayer force continues to rise to a value of 1.5kN while the faller force drops to 1.5kN and the measured top runner force is down to 3.1kN which also adds up nicely. All well and good.

However if you are going to use the maximum forces and give a factor to convert then it all goes a bit haywire, here are the measured forces for two common plates, one a soft one and one more powerful.

So... it's complicated (read: more or less impossible) to calculate from first principles..... the only way to get good estimates of forces is by experiments with various configurations.

This isn't true if by first principles you mean starting with the same assumptions you did. It is relatively straightforward to calculate Ft while including f as a function of theta.

As theta varies up to a pretty large angle the difference in force is small enough to be considered noise, as you've already admitted. And of course Jim Titt pointed out there are much larger real effects that are being ignored by this "first principles" approach.

But that doesn't mean that real forces cannot be estimated with models.

So... it's complicated (read: more or less impossible) to calculate from first principles..... the only way to get good estimates of forces is by experiments with various configurations.

This isn't true if by first principles you mean starting with the same assumptions you did. It is relatively straightforward to calculate Ft while including f as a function of theta.

As theta varies up to a pretty large angle the difference in force is small enough to be considered noise, as you've already admitted. And of course Jim Titt pointed out there are much larger real effects that are being ignored by this "first principles" approach.

But that doesn't mean that real forces cannot be estimated with models.

The difficulty lies in predicting with any accuracy how f varies, and all the other unforeseen factors such as the (very interesting) hysteresis effects Jim pointed out.

But that's my point, really. You could put together a model entirely from first principles, but it would be really quite difficult and most likely a fair way off from reality. Easier to collect a bunch of experimental data to feed into a semi-empirical model.

But that's my point, really. You could put together a model entirely from first principles, but it would be really quite difficult and most likely a fair way off from reality. Easier to collect a bunch of experimental data to feed into a semi-empirical model.

If that were really your original point it would have been better had you simply stated that in the first place. It would have led to a different and possibly less trivial discussion.

It is when you start to consider factors such as rope age, stiffness and fuzziness, carabiner model, wetness, etc.

In reply to:

In reply to:

But that's my point, really. You could put together a model entirely from first principles, but it would be really quite difficult and most likely a fair way off from reality. Easier to collect a bunch of experimental data to feed into a semi-empirical model.

If that were really your original point it would have been better had you simply stated that in the first place. It would have led to a different and possibly less trivial discussion.

The angle thing is sort-of interesting but believe me to derive anything from first principles appears almost impossible, at some stage you have to plug some experimental data in somewhere and to get the data youīve effectively already done the tests and got the answers!

The classic way to work out the friction around the karabiner is to plug the coefficient of friction and angle into Amontons capstan theory but this doesnīt work for ropes which have a significant bending resistance (stiffness) and capstan theory is a dead duck here. The frictional force varies with angle, rope stiffness and load and to calculate the rope stiffness is going to be a monster job with thousands of individual threads all moving around under load. And that is if the coefficient of friction is a constant, since however it varies with pressure and speed amongst other things this is another load of variables for each different case. Going to be a complicated formula!

Obviously its going to be easier to just bend it around something and measure at various loads and angles which is what we wanted anyway. Iīve already done this for lower loads (that occur in the belay device) and itīs clear that after a wrap angle of 90° the increase in frictional force is levelling off and by 180° there is virtually no increase. (Technically it isnīt frictional force but pulley efficiency as friction is not the only effect).

To test at the higher loads encountered in a fall is easy enough but practically testing at the speeds and loads together is going to be another matter. Since the normal drop tests (which are normally conducted at 150° to stop the drop-weight landing on the belayer below) are about the worst case scenario it seems reasonable to use these results.